The first class includes the initial-state radiation of low-virtuality vector bosons. It effectively makes the muon collider a high-luminosity vector bosons collider, on top of a very high-energy lepton-lepton machine. The compelling associated physics studies described in Section  pose challenges for fixed-order theoretical predictions and Monte Carlo event generation even at tree-level, owing to the sharp features of the Monte Carlo integrand induced by the large scale separation and the need to correctly handle QED and weak radiation at the same time, respecting EW gauge invariance. Strategies to address these challenges are available in WHIZARD [1]}, they have been recently implemented in MadGraph5_aMC@NLO [2]}, [3]} and applied to several phenomenological studies in the muon collider context. Dominance of such initial-state collinear radiation will eventually require a systematic theoretical modeling in terms of EW Parton Distribution Function where multiple collinear radiation effects are resummed. First studies show that EW resummation effects can be significant at a 10 TeV muon collider [4]}.
